We study the problem of minimizing the diameter of a graph by adding k
shortcut edges, for speeding up communication in an existing network design.
We develop constant-factor approximation algorithms for different variations
of this problem. We also show how to improve the approximation ratios using
resource augmentation to allow more than k shortcut edges. We observe
a close relation between the single-source version of the problem, where we
want to minimize the largest distance from a given source vertex, and the
well-known k-median problem. First we show that our constant-factor
approximation algorithms for the general case solve the single-source problem
within a constant factor. Then, using a linear-programming formulation for
the single-source version, we find a (1 + ε)-approximation
using O(k log n) shortcut edges. To show the
tightness of our result, we prove that any
(3/2 − ε)-approximation for the single-source version
must use Ω(k log n) shortcut edges assuming
P ≠ NP.